The CENTRE for EDUCATION in MATHEMATICS and COMPUTING

cemc.uwaterloo.ca

## Canadian Senior

## Mathematics Contest

Thursday, November 20, 2014 (in North America and South America)

Friday, November 21, 2014

(outside of North America and South America)

Time: 2 hours _{}c2014 University of Waterloo

Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet.

Do not open this booklet until instructed to do so.

There are two parts to this paper. The questions in each part are arranged roughly in order of increasing difficulty. The early problems in Part B are likely easier than the later problems in Part A.

PART A

1. This part consists of six questions, each worth 5 marks.

2. Enter the answer in the appropriate box in the answer booklet.

For these questions, full marks will be given for a correct answer which is placed in the box. Part marks will be awardedonly if relevant workis shown in the space provided in the answer booklet.

PART B

1. This part consists of three questions, each worth 10 marks.

2. Finished solutions must be written in the appropriate location in the answer booklet. Rough work should be done separately. If you require extra pages for your finished solutions, they will be supplied by your supervising teacher. Insert these pages into your answer booklet. Be sure to write your name, school name and question number on any inserted pages.

3. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

At the completion of the contest, insert your student information form inside your answer booklet.

Do not discuss the problems or solutions from this contest online for the next 48 hours.

### Canadian Senior Mathematics Contest

NOTE:

1. Please read the instructions on the front cover of this booklet.

2. Write solutions in the answer booklet provided.

3. It is expected that all calculations and answers will be expressed as exact

numbers such as 4π, 2 +√7, etc., rather than as 12.566. . .or 4.646. . ..

4. While calculators may be used for numerical calculations, other mathematical

steps must be shown and justified in your written solutions and specific marks may be allocated for these steps. For example, while your calculator might be

able to find the x-intercepts of the graph of an equation likey=x3_{−}x, you

should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

5. Diagrams are not drawn to scale. They are intended as aids only.

6. No student may write both the Canadian Senior Mathematics Contest and the

Canadian Intermediate Mathematics Contest in the same year.

### PART A

For each question in Part A, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet.

1. In the diagram, ABCD is a square, _{△}ABE is

equilateral, and _{△}AEF is equilateral. What is the

measure of∠DAF? total value of these coins is $4, how many dimes are in the jar?

(Each dime is worth 10 cents, each quarter is worth 25 cents, and $1 equals 100 cents.)

3. Positive integers m and n satisfy mn = 5000. If m is not divisible by 10 and n is

not divisible by 10, what is the value of m+n?

4. A function f satisfiesf(x) +f(x+ 3) = 2x+ 5 for all x.

Iff(8) +f(2) = 12, determine the value off(5).

5. Determine all real numbersx for which p3

(2 +x)2_{+ 3}p3

(2_{−}x)2 _{= 4}√3

4_{−}x2_{.}

6. Ten lockers are in a row. The lockers are numbered in order with the positive integers 1 to 10. Each locker is to be painted either blue, red or green subject to the following rules:

• Two lockers numbered m and n are painted different colours whenever m_{−}n

is odd.

• It is not required that all 3 colours be used.

### PART B

For each question in Part B, your solution must be well organized and contain words of explanation or justification. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

1. An arithmetic sequence is a sequence in which each term after the first is obtained

from the previous term by adding a constant. For example, 5,7,9 is an arithmetic

sequence with three terms.

(a) Three of the five numbers 2,5,10,13,15 can be chosen to form an arithmetic

sequence with three terms. What are the three numbers?

(b) The numbers p,7, q,13, in that order, form an arithmetic sequence with four

terms. Determine the value of pand the value of q.

(c) The numbers a, b, c,(a+ 21), in that order, form an arithmetic sequence with

four terms. Determine the value ofc_{−}a.

(d) The numbers (y_{−}6),(2y+3),(y2+2), in that order, form an arithmetic sequence

with three terms. Determine all possible values ofy.

2. (a) (i) In the diagram, semi-circles are drawn on the

sides of right-angled _{△}XY Z, as shown. If

the area of the semi-circle with diameterY Z

is 50π and the area of the semi-circle with

diameter XZ is 288π, determine the area of

the semi-circle with diameterXY.

*X* _{Y}

*Z*

*x*
*y*

*z*

(ii) In the diagram, squares with side

lengths p,q and r are constructed on

the sides of right-angled _{△}P QR, as

shown. Diagonals QT, RU and P S

have lengths a, b and c, respectively.

Prove that the triangle formed with

sides of lengths a, b and c is a

right-angled triangle. *P* *Q*

*R*

(b) In the diagram,_{△}XY Z is right-angled at

Z and AZ is the altitude fromZ toXY.

Also, segmentsADandAB are altitudes

in_{△}AXZ and_{△}AY Z, respectively, and

segments DE and BC are altitudes in

△ADX and _{△}ABY, respectively. Prove

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3. For any real number x, _{⌊}x_{⌋} denotes the largest integer less than or equal to x.

For example, _{⌊}4.2_{⌋} = 4 and _{⌊−}2.4_{⌋} =_{−}3. That is, _{⌊}x_{⌋} is the integer that satisfies

the inequality_{⌊}x_{⌋ ≤}x <_{⌊}x_{⌋}+ 1.

(a) The equation x2 = 3_{⌊}x_{⌋}+ 1 has two solutions. One solution is x = √7. The

second solution is of the form x =√afor some positive integer a. Determine

the value ofa.

(b) For each positive integern, determine all possible integer values of the expression

x2_{−}3_{⌊}x_{⌋}, where x is a real number with_{⌊}x_{⌋}=n.

(c) For each integer k with k _{≥} 0, determine all real numbers x for which